Find the inverse of the function:
f(t) = (10^t - 10^(-t))/(10^t + 10 ^ (-t))
Thanks,
Anshu
$\displaystyle f(t)=y=\dfrac{10^t-\dfrac1{10^t}}{10^t+\dfrac1{10^t}} = \dfrac{10^{2t}-1}{10^{2t}+1}$
Solve the equation for t:
$\displaystyle y= \dfrac{10^{2t}-1}{10^{2t}+1}$ Substitute $\displaystyle a = 10^{2t}$
$\displaystyle y= \dfrac{a-1}{a+1}~\implies~a=\dfrac{y+1}{y-1}$
Now re-substitute:
$\displaystyle 10^{2t}=\dfrac{y+1}{y-1}~\implies~2t=\log\left(\dfrac{y+1}{y-1}\right)~\implies~\boxed{t=\dfrac12\cdot \log\left(\dfrac{y+1}{y-1}\right)}$